The Hubble Tension from S² Geometry: A Zero-Parameter Resolution via a Time-Varying Hubble Function H(θ) = H_int sin(2θ)
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2026
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| author | de Haan, C. R. (René) |
| author_facet | de Haan, C. R. (René) |
| contents | <p class="MsoNormal">The Hubble tension — the ~5σ disagreement between H₀ measured from the cosmic microwave background (Planck 2018: 67.4 ± 0.5 km/s/Mpc) and from local standard candles (SH0ES 2022: 73.0 ± 1.0 km/s/Mpc) — is one of the most significant unresolved problems in cosmology. Most proposed resolutions either modify early-universe physics (early dark energy, extra relativistic species) or modify the late-universe expansion (interacting dark energy, local voids), at the cost of introducing one or more free parameters.</p> <p class="MsoNormal">We present a resolution in the Three Time Dimensions (3+3) spacetime framework with zero adjustable parameters, in which the Hubble function is not constant but a geometric function of the cosmic angle θ:</p> <p class="MsoNormal">H(θ) = H_int × sin(2θ)</p> <p class="MsoNormal">with H_int = 78.175 km/s/Mpc derived from S² compactification geometry (0.014% from its self-consistent value) and θ the t₂ precession angle that advances from 0 at the Bang to π/2 at maximum expansion. This single formula replaces the ΛCDM Friedmann equation. At the CMB decoupling epoch θ ≈ 0.04°, the observed H value corresponds to a different point on the sin(2θ) curve than at the current epoch θ = 55.79° — so CMB and local measurements are measuring different things rather than contradicting each other.</p> <p class="MsoNormal">The prediction at the current epoch is:</p> <p class="MsoNormal">H₀ = H_int × sin(2θ_now) = 72.7 km/s/Mpc</p> <p class="MsoNormal">The prediction falls between the Planck CMB value (67.4, 7.3% below) and the SH0ES local value (73.0, 0.4% below). The derivation of H_int uses the same geometric structure — a double projection at the magic angle θ_node = arccos(1/√3) = 54.74°, yielding sin⁴(θ_node) = 4/9 — that appears in the zero-parameter derivation of the fine-structure constant α = 1/137.036 within the same framework. The fine-structure constant and the Hubble constant share one geometric origin.</p> <p class="MsoNormal">The framework makes a second, sharper prediction: the dark-energy equation of state w(θ) = −1 + (2/3)(θ̇/H(θ)) cot(2θ) crosses w = −1 exactly at the geometric epoch θ = 45°, corresponding to redshift z = 0.223. The prediction is that w(z) shows phantom behaviour (w < −1) at z < 0.22 and quintessence behaviour (w > −1) at z > 0.22, with a sharp shape change around the crossing. This is not a CPL curve, and all six DESI 2024 BAO tracers currently agree with the prediction at <0.2σ. DESI DR2 (expected 2026) will test the crossing directly at z ≈ 0.2–0.3.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19666441 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Hubble Tension from S² Geometry: A Zero-Parameter Resolution via a Time-Varying Hubble Function H(θ) = H_int sin(2θ) de Haan, C. R. (René) Hubble tension Hubble constant H₀ dark energy equation of state w(z) DESI SH0ES Planck CMB (3+3) spacetime three time dimensions S² compactification Kaluza-Klein magic angle cosmic evolution zero-parameter cosmology cosmological constant dark energy equation of state w=-1 crossing BAO phantom dark energy quintessence Friedman equation late-universe cosmology <p class="MsoNormal">The Hubble tension — the ~5σ disagreement between H₀ measured from the cosmic microwave background (Planck 2018: 67.4 ± 0.5 km/s/Mpc) and from local standard candles (SH0ES 2022: 73.0 ± 1.0 km/s/Mpc) — is one of the most significant unresolved problems in cosmology. Most proposed resolutions either modify early-universe physics (early dark energy, extra relativistic species) or modify the late-universe expansion (interacting dark energy, local voids), at the cost of introducing one or more free parameters.</p> <p class="MsoNormal">We present a resolution in the Three Time Dimensions (3+3) spacetime framework with zero adjustable parameters, in which the Hubble function is not constant but a geometric function of the cosmic angle θ:</p> <p class="MsoNormal">H(θ) = H_int × sin(2θ)</p> <p class="MsoNormal">with H_int = 78.175 km/s/Mpc derived from S² compactification geometry (0.014% from its self-consistent value) and θ the t₂ precession angle that advances from 0 at the Bang to π/2 at maximum expansion. This single formula replaces the ΛCDM Friedmann equation. At the CMB decoupling epoch θ ≈ 0.04°, the observed H value corresponds to a different point on the sin(2θ) curve than at the current epoch θ = 55.79° — so CMB and local measurements are measuring different things rather than contradicting each other.</p> <p class="MsoNormal">The prediction at the current epoch is:</p> <p class="MsoNormal">H₀ = H_int × sin(2θ_now) = 72.7 km/s/Mpc</p> <p class="MsoNormal">The prediction falls between the Planck CMB value (67.4, 7.3% below) and the SH0ES local value (73.0, 0.4% below). The derivation of H_int uses the same geometric structure — a double projection at the magic angle θ_node = arccos(1/√3) = 54.74°, yielding sin⁴(θ_node) = 4/9 — that appears in the zero-parameter derivation of the fine-structure constant α = 1/137.036 within the same framework. The fine-structure constant and the Hubble constant share one geometric origin.</p> <p class="MsoNormal">The framework makes a second, sharper prediction: the dark-energy equation of state w(θ) = −1 + (2/3)(θ̇/H(θ)) cot(2θ) crosses w = −1 exactly at the geometric epoch θ = 45°, corresponding to redshift z = 0.223. The prediction is that w(z) shows phantom behaviour (w < −1) at z < 0.22 and quintessence behaviour (w > −1) at z > 0.22, with a sharp shape change around the crossing. This is not a CPL curve, and all six DESI 2024 BAO tracers currently agree with the prediction at <0.2σ. DESI DR2 (expected 2026) will test the crossing directly at z ≈ 0.2–0.3.</p> |
| title | The Hubble Tension from S² Geometry: A Zero-Parameter Resolution via a Time-Varying Hubble Function H(θ) = H_int sin(2θ) |
| topic | Hubble tension Hubble constant H₀ dark energy equation of state w(z) DESI SH0ES Planck CMB (3+3) spacetime three time dimensions S² compactification Kaluza-Klein magic angle cosmic evolution zero-parameter cosmology cosmological constant dark energy equation of state w=-1 crossing BAO phantom dark energy quintessence Friedman equation late-universe cosmology |
| url | https://doi.org/10.5281/zenodo.19666441 |